Research on a Chassis Stability Control Method for High-Ground-Clearance Self-Propelled Electric Sprayers

Abstract

In response to the complex working conditions and poor driving stability of high-clearance self-propelled sprayers, a nonlinear model of the chassis power system was established based on the independently controllable torque of each wheel of the developed electric sprayer. A layered-architecture chassis drive control strategy was formulated, and a stability control framework comprising an instability judgment module, an upper controller, and a lower controller was constructed based on the analysis of the impact of the centroid slip angle, the yaw rate, and the wheel slip rate on driving stability. An ideal reference model was established based on the seven-degree-of-freedom model of the sprayer, and the current state of the sprayer body was determined using the instability judgment module. A drive anti-slip controller and a yaw moment controller based on fuzzy PID theory and sliding mode control theory were designed. Additionally, an optimal torque distribution algorithm was developed based on tire characteristics to rationally allocate drive torque to each wheel, ensuring the stability of the sprayer during operation. Simulation tests were conducted using MATLAB/Simulink to evaluate the sprayer under four different driving conditions during transport and field operations. The test results showed that the “SMC + optimal distribution” control method in the chassis stability control strategy reduced the maximum deviations of the yaw rate and centroid slip angle by an average of 89.5% and 13.6%, respectively, compared to no control. The wheel slip rate during straight driving was well maintained at around 15%, enhancing the driving stability of the sprayer.

1. Introduction

The high-clearance self-propelled electric sprayer, characterized by its substantial ground clearance, demonstrates broad adaptability to plant protection operations in tall-stalk crops, thereby holding immense potential for widespread application [1,2]. Currently, the majority of high-clearance self-propelled sprayer chassis are hydraulically driven and equipped with hydraulic power steering systems [3,4,5,6]. However, in recent years, advancements in motor and electronic control technologies have led to the increasing adoption of distributed chassis structures equipped with four in-wheel motors in agricultural machinery chassis. This configuration enables the independent control of the output torque of each drive motor, effectively mitigating issues such as mudding and inadequate stability that have traditionally plagued high-clearance sprayers during operation [7,8,9,10]. Nonetheless, this distributed drive layout also complicates the joint control of the vehicle chassis’s dynamic performance, escalating the challenge of system control [11]. Consequently, ensuring the control precision and managing the control complexity for maintaining the stability of the sprayer chassis have emerged as pressing issues requiring prompt resolution.

Currently, research by scholars both domestically and internationally on the stability control of four-wheel-drive vehicles mainly focuses on direct yaw moment control [12,13,14], drive torque distribution technology [15,16], and electronic differential control [17]. Additionally, for the sprayer chassis system, which exhibits strong nonlinear characteristics, the choice of an appropriate stability control algorithm is crucial. Currently, the model predictive control [18], sliding-mode variable structure control [19], fuzzy control [20], or a combination of multiple control algorithms [21] are commonly used.

However, the current research on the stability control of sprayers is mostly limited to the control of a single index such as the wheel slip rate or lateral or longitudinal stability, and an in-depth discussion on the cooperative control strategy of multiple stability indicators is lacking. At the same time, a control strategy combined with the specific working scene of the sprayer is relatively scarce. In view of this, this study aims to fill the research gap by combining the structural characteristics and operational requirements of a self-propelled sprayer and proposing an innovative chassis control strategy, aiming to optimize the side deflection angle of the center of mass, yaw speed, and the wheel slip rate at the same time, so as to maximize the stability of the whole machine. Through the carefully designed torque distribution algorithm, the reasonable distribution of each wheel torque can be ensured to improve the performance and stability of the sprayer under various working conditions. Finally, the controller simulation model built in the MATLAB/Simulink (R2023a) environment fully demonstrates the effectiveness and practicability of the innovative control strategy through the simulation verification of four driving situations under different working conditions.

2. Materials and Methods

2.1. Establishment of a Dynamic Model for the Sprayer Chassis

An ideal model is crucial in the stability control of sprayers. It allows for the determination of the ideal yaw rate and the sideslip angle of the center of mass based on the current wheel angle. By analyzing the sprayer’s coordinate system (Figure 1), a two-degree-of-freedom model can be established to analyze the ideal steering characteristics of the sprayer. This study solely focused on the control strategy for chassis stability during front-wheel steering; hence, in Figure 1, only the two front wheels of the sprayer are depicted as participating in the steering process.

A two-degree-of-freedom model for the sprayer chassis was established based on the following assumptions:

(1) The model input was the front wheel steering angle; (2) the effect of the suspension was neglected; (3) the vehicle’s forward speed was constant; (4) the driving force was minimal; (5) the effect of air resistance was ignored; and (6) the mass and inertia of the wheel were ignored.

Based on the above assumptions, the two-degree-of-freedom model of the sprayer could describe the vehicle’s motion on a plane, as shown in Figure 2, including the translational motion and rotational motion along the vertical axis.

According to Figure 2, the dynamic equations of the lateral and yaw of the sprayer, the lateral forces F_{y_f} and F_{y_r} of the front and rear wheels, and the side deflection angle α_f and α_r can be derived, which are specifically expressed using Equation (1) [22] as follows:

$$\left\{\begin{array}{l}M{v}_{\mathrm{x}}({\beta }_{\mathrm{centroid}}^{’}+{\omega }_{\mathrm{z}})=F_{\mathrm{y}\_\mathrm{f}}\cos {\delta }_{\mathrm{f}}+F_{\mathrm{y}\_\mathrm{r}}\\ I_{\mathrm{z}}{\omega }_{\mathrm{z}}=l_{\mathrm{b}}{F}_{\mathrm{y}\_\mathrm{f}}\cos {\delta }_{\mathrm{f}}-l_{\mathrm{a}}{F}_{\mathrm{y}\_\mathrm{r}}\\ F_{\mathrm{y}\_\mathrm{f}}=k_{\mathrm{f}}\cdot {\alpha }_{\mathrm{f}}\\ F_{\mathrm{y}\_\mathrm{r}}=k_{\mathrm{r}}\cdot {\alpha }_{\mathrm{r}}\\ {\alpha }_{\mathrm{f}}={\beta }_{\mathrm{centroid}}-{\delta }_{\mathrm{f}}+{\displaystyle \frac{l_{\mathrm{a}}{\omega }_{\mathrm{z}}}{v_{\mathrm{x}}}}\\ {\alpha }_{\mathrm{r}}={\beta }_{\mathrm{centroid}}-{\displaystyle \frac{l_{\mathrm{b}}{\omega }_{\mathrm{z}}}{v_{\mathrm{x}}}} \end{array}\right.$$

(1)

In Equation (1), k_f and k_r represent the lateral stiffness of the front and rear axle (sum of the lateral stiffness of the two wheels) respectively, N/rad.

Assuming that the angle of the front wheel is small, sinδ_f ≈ δ_f and cosδ_f ≈ 1 are satisfied. After solving Equation (1), the state equation of the two-degree-of-freedom spray mechanical model is obtained, as shown in Equation (2).

$$\left[\begin{array}{c}{\beta }_{\mathrm{centroid}}^{’}\\ {\omega }_{\mathrm{z}}^{’}\end{array}\right]=\left[\begin{array}{cc}{\displaystyle \frac{k_{\mathrm{f}}+k_{\mathrm{r}}}{M{v}_{\mathrm{x}}}}& {\displaystyle \frac{l_{\mathrm{a}}{k}_{\mathrm{f}}-l_{\mathrm{b}}{k}_{\mathrm{r}}}{M{v}_{\mathrm{x}}^2}}-1\\ {\displaystyle \frac{l_{\mathrm{a}}{k}_{\mathrm{f}}-l_{\mathrm{b}}{k}_{\mathrm{r}}}{I_{\mathrm{z}}}}& {\displaystyle \frac{l_{\mathrm{a}}^2{k}_{\mathrm{f}}+l_{\mathrm{b}}^2{k}_{\mathrm{r}}}{I_{\mathrm{z}}{v}_{\mathrm{x}}}}\end{array}\right]\left[\begin{array}{c}{\beta }_{\mathrm{centroid}}\\ {\omega }_{\mathrm{z}}\end{array}\right]+\left[\begin{array}{c}{\displaystyle \frac{-k_{\mathrm{f}}}{M{v}_{\mathrm{x}}}}\\ {\displaystyle \frac{-l_{\mathrm{a}}{k}_{\mathrm{f}}}{I_{\mathrm{z}}}}\end{array}\right]{\delta }_{\mathrm{f}}$$

(2)

For Equation (2), when the vehicle reaches a steady state, the centroid slip angle and the yaw rate become constants, with their derivatives ${\beta }_{\mathrm{centroid}}^{’}$ and ${\omega }_z^{’}$ equal to zero. At this point, the desired yaw rate can be derived as follows [23]:

$$\left\{\begin{array}{l}{\omega }_{\mathrm{z}\_\exp }={\displaystyle \frac{v_{\mathrm{x}}{\delta }_{\mathrm{f}}}{L\left(1+K{v}_{\mathrm{x}}^2\right)}}\\ K={\displaystyle \frac{M}{L^2}}\left({\displaystyle \frac{l_{\mathrm{b}}}{k_{\mathrm{r}}}}-{\displaystyle \frac{l_{\mathrm{a}}}{k_{\mathrm{f}}}}\right)\end{array}\right.$$

(3)

In Equation (3), K represents the stability factor. When K = 0, the yaw rate ω is directly proportional to the vehicle speed v_x, indicating a neutral steering state of the sprayer. When K < 0, the sprayer exhibits oversteering; when K > 0, it exhibits understeering. Both oversteering and understeering can lead to a loss of steering stability.

Due to the limitations of ground adhesion, the acceleration at the vehicle’s centroid must not exceed the maximum allowable acceleration of the road surface. Since the lateral speed is relatively small compared to the longitudinal speed, it can be neglected. Additionally, a stability factor λ is introduced to enhance steering stability. The maximum desired yaw moment can thus be expressed as follows [24]:

$$\left|{\omega }_{\mathrm{z}}\right|\le \lambda \left|\frac{\mu g}{v_{\mathrm{x}}}\right|$$

(4)

In Equation (4), $\lambda$ is the stability factor set to 0.9, and μ is the road adhesion coefficient. Ideally, the centroid slip angle of the sprayer should be minimized to zero to maintain the accuracy and stability of the vehicle’s trajectory.

In summary, the final desired yaw rate and the ideal value of the centroid slip angle are determined using Equation (5) as follows:

$$\left\{\begin{array}{c}{\omega }_{\mathrm{z}\_\exp }=min\left({\displaystyle \frac{v_{\mathrm{x}}{\delta }_{\mathrm{f}}}{L\left(1+K{v}_{\mathrm{x}}^2\right)}},\lambda \left|{\displaystyle \frac{\mu g}{v_{\mathrm{x}}}}\right|\right)\\ {\beta }_{\mathrm{z}\_\exp }=0\end{array}\right.$$

(5)

Based on the ideal model, a seven-degree-of-freedom nonlinear steering model, including longitudinal, lateral, yaw, and four-wheel rotational motions, was established [25]. Finally, a corresponding model was developed in MATLAB/Simulink using the actual parameters of the sprayer. The main parameters of the sprayer are listed in

Table 1. Results of the spray boom overall height adjustment test.

Parameter Name	Parameter Unit	Description
Chassis drive form	/	Four-wheel independent motor drive
Steering form	/	Four-wheel independent motor steering
Vehicle full load mass	kg	450
Distance from center of mass to front axis	mm	900
Distance from center of mass to rear axis	mm	700
Sprayer wheelbase	mm	1200
Ground clearance	mm	600
Maximum working speed	km/h	20
Minimum working speed	km/h	3

.

2.2. Analysis of Chassis Stability Architecture

We used the yaw rate, centroid slip angle, and wheel slip rate as control variables. A combination of the sliding mode control algorithm and the fuzzy PID control algorithm served as the main framework for the drive coordination control strategy. Additionally, a speed-following control was integrated to enable the sprayer to calculate the required drive torque and adjust the yaw moment based on control commands, ensuring that the wheel slip rate would not exceed the limit. Finally, the torque optimal distribution algorithm allocated torque to each wheel, resulting in a driving stability control system for the sprayer. This system ensured that the sprayer would closely approach or even reach the desired state to maintain stable driving.

The overall control process for the sprayer chassis stability control strategy studied in this paper was as follows:

(1)

Determine the vehicle stability based on control signal commands and the current operating state of the sprayer to decide whether stability control intervention is needed.

(2)

If the vehicle is deemed stable, perform speed-following control alone by adjusting the sprayer’s acceleration through the speed-following controller to match the desired speed. If instability is detected, calculate the expected stability-related parameters from the ideal two-degree-of-freedom model (the centroid slip angle and the yaw rate) and determine the deviation from actual parameters. Adjust the drive torque of each wheel based on the deviation to generate a yaw moment that ensures stable driving. Simultaneously, monitor the wheel slip rate, and if any wheel exceeds the acceptable slip rate range, correct the torque of the affected wheel to prevent the loss of directional control.

(3)

Finally, use the allocation algorithm of the lower-level controller to reasonably distribute the longitudinal drive torque and the adjustment torque output by the upper-level controller to each wheel, thereby achieving the stability control of the sprayer chassis.

Based on the above analysis, the chassis stability control strategy of this paper is shown in Figure 3. The stability control system mainly consisted of a two-degree-of-freedom reference model, an instability judgment module, an upper-layer torque control module, and a lower-layer torque distribution module. After determining the overall architecture of the chassis stability control strategy, a detailed analysis of the entire system yielded the refined stability control strategy shown in Figure 4.

2.3. Sprayer’s Unstable-State Judgment

Instability mainly occurs during steering, with common types being understeering and oversteering. Additionally, high wheel slip rates can lead to dangerous situations like fishtailing, which alters the vehicle’s posture, causing significant deviations in the centroid slip angle or yaw rate. Therefore, the simultaneous monitoring of the yaw rate and centroid slip angle is necessary to determine if the sprayer is unstable.

Based on Guo Jianhua’s research on the dβ-β phase plane [26], we employed a method that integrates stability parameters with the yaw rate deviation threshold for comprehensive instability assessment. The specific methods and procedures for instability determination are as follows:

(1)

Calculate the yaw rate deviation Δω and the centroid slip angle deviation Δβ_centroid;

(2)

Determine parameters B₁ and B₂ from

Table 2. Stability determination parameters.

The Value of the Road Adhesion Coefficient μ Is Taken	B1	B2
0–0.2	0.284	2.577
0.2–0.4	0.297	3.345
0.4–0.6	0.303	4.228
0.6–0.8	0.357	4.654
0.8–1	0.357	5.573

according to the road adhesion coefficient and assess the sprayer’s stability using Equation (6) as follows:

$$\left|{\beta }_{\mathrm{centroid}}+B_1{\dot{\beta }}_{\mathrm{centroid}}\right|\le B_2$$

(6)

(3)

If Equation (6) is not satisfied, it indicates that the sprayer is unstable, necessitating stability control; if satisfied, further evaluate whether the yaw rate deviation K is within the critical value range based on the current speed, with critical values given in

Table 3. The critical value of yaw rate deviation K in the instability zone.

Speed (km/h)	Critical Value of the Lateral Angular Speed Deviation K for the Unstable Zone
5	0.0195
10	0.0200
15	0.0205
20	0.0210
25	0.0215

;

(4)

If the critical value is exceeded, stability control is required due to excessive yaw rate deviation; otherwise, the sprayer is considered stable.

2.4. Upper-Stability Controller Design

2.4.1. Design of ASR-Driven Anti-Slip Controller

The anti-slip drive controller utilizes a fuzzy PID control method. Its primary control objective is to reduce the slip rate of the wheels when the controller detects that it exceeds a threshold, keeping it near the target slip rate.

Generally, the optimal slip rate of the wheels ranges from 10% to 20%, ensuring sufficient friction between the wheels and the ground to provide effective traction and braking performance [27,28]. In this study, the optimal slip rate was set at 0.15, ensuring that both longitudinal and lateral adhesion coefficients of the wheels remained high. When the calculated slip rate of the wheels exceeded the threshold of 0.15, the deviation from the target value and its rate of change were computed. The control algorithm then determined the correction value ΔT for the anti-slip torque of each wheel, ensuring that the wheel slip rate remained within an appropriate range.

Calculation of the Wheel Slip Rate

The slip rates of the wheels, denoted as λ_wij, are determined as follows:

$${\lambda }_{\mathrm{wij}}={\displaystyle \frac{{\omega }_{\mathrm{wij}}R-v_{\mathrm{xij}}}{\max ({\omega }_{\mathrm{wij}}R,v_{\mathrm{xij}})}}$$

(7)

In Equation (7), v_xij represents the velocity of each wheel in the longitudinal axis direction, where i = f, r; j = l. r; when the slip rate λ_wij > 0, the wheel is in a slip condition; when the slip rate λ_wij < 0, the wheel is in a skidding condition.

Design of Fuzzy PID Controller

The fuzzy PID control algorithm employs the principles of fuzzy mathematics to formulate fuzzy rules, which, along with the initial PID control parameters, are stored in a knowledge base. The controller responds to actual parameter changes, offering robust, and widely applicable characteristics [29].

Fuzzy PID was used for active control of the ASR drive anti-slip system. In order to ensure that the slip rate of each wheel would fall in a stable region, the fuzzy domain was divided into [−3, −2, −1, 0, 1, 2, 3] based on the actual range of slip rate deviation and its change rate and combined with the accuracy requirements of the control system. At the same time, in order to describe the different degrees of slip rate deviation and the change rate more carefully, so as to improve the sensitivity and accuracy of the control system, seven fuzzy subsets were defined. These subsets were negative big (NB), negative medium (NM), negative small (NS), zero (ZO), positive small (PS), positive medium (PM), and positive big (PB). The output was the torque adjustment for the wheels, with its fuzzy domain and subsets aligned with the input quantities. The membership functions were triangular in the middle and Gaussian on the sides. This ensures the control smoothness, and the responseability of the system is enhanced under extreme conditions. The membership functions for the input and output quantities are shown in Figure 5.

Based on the principles of fuzzy rule establishment, fuzzy rules were formulated according to the changes in wheel slip rates. The final fuzzy control rules are shown in

Table 4. ΔK_p, ΔK_i, ΔK_d, and fuzzy PID control rules.

e	ec
	NB	NM	NS	ZO	PS	PM	PB
NB	PB/NB/PS	PB/NB/NS	PM/NM/NB	PM/NM/NB	PS/NS/NB	ZO/ZO/NM	ZO/ZO/ZO
NM	PB/NB/PS	PM/NB/NS	PM/NM/NB	PS/NS/NM	PS/NS/NM	ZO/ZO/NS	NS/ZO/ZO
NS	PM/NB/ZO	PM/NM/NS	PS/NS/NM	PS/NS/NM	ZO/ZO/NS	NS/PS/NS	NM/PS/ZO
ZO	PM/NM/ZO	PM/NM/NS	PS/NS/NS	ZO/ZO/NS	NS/PS/NS	NM/PM/NS	NM/PM/ZO
PS	PM/NM/ZO	PS/NS/ZO	ZO/ZO/ZO	NS/PS/ZO	NM/PS/ZO	NM/PM/ZO	NM/PB/ZO
PM	PS/ZO/PB	ZO/ZO/NS	NS/PS/PS	NS/PS/PS	NM/PM/PS	NB/PB/PS	NB/PB/PB
PB	ZO/ZO/PB	ZO/ZO/PM	NM/PS/PM	NM/PM/PM	NM/PM/PS	NB/PB/PS	NB/PB/PB

.

The fuzzy rule controller surface, derived from the fuzzy rules, is shown in Figure 6.

The controller model was established in MATLAB/Simulink. The controller for a single wheel is shown in Figure 7, and the slip rate control module structure is shown in Figure 8. The model’s inputs were the wheel slip ratio and the slip ratio error, while the output was the corrective torque. The controller’s proportional factors 1 and 2 were set to 3/0.16 and 3/0.5, respectively. The individual quantization factors were set to 1/3, 0.05/3, and 0.01/3. The basic control parameters for the fuzzy PID were a proportional coefficient of 10, an integral coefficient of 0.5, and a derivative coefficient of 0.1.

2.4.2. Design of Direct Yaw Moment Controller

When the sprayer turns, the direct yaw moment controller identifies the sprayer’s driving intention based on the steering angle signal. Utilizing the previously established chassis dynamics model, the required yaw moment for the sprayer is calculated through the corresponding algorithm. This ensures the sprayer maintains neutral steering to the greatest extent possible, enhancing its driving stability under extreme conditions. The specific control system structure is shown in Figure 9.

By incorporating an additional yaw moment ΔT into the two-degree-of-freedom sprayer dynamics model, the new dynamic equation is obtained as follows:

$$\left\{\begin{array}{c}{\dot{\beta }}_{\mathrm{centroid}}={\displaystyle \frac{k_{\mathrm{f}}+k_{\mathrm{r}}}{M{v}_{\mathrm{x}}}}{\beta }_{\mathrm{centroid}}+\left({\displaystyle \frac{l_{\mathrm{a}}{k}_{\mathrm{f}}-l_{\mathrm{b}}{k}_{\mathrm{r}}}{M{v}_{\mathrm{x}}^2}}-1\right){\omega }_{\mathrm{z}}-{\displaystyle \frac{k_{\mathrm{f}}}{M{v}_{\mathrm{x}}}}{\delta }_{\mathrm{f}}\\ {\dot{\omega }}_{\mathrm{z}}={\displaystyle \frac{l_{\mathrm{a}}{k}_{\mathrm{f}}-l_{\mathrm{b}}{k}_{\mathrm{r}}}{I_{\mathrm{z}}}}{\beta }_{\mathrm{centroid}}+{\displaystyle \frac{l_{\mathrm{a}}^2{k}_{\mathrm{f}}+l_{\mathrm{b}}^2{k}_{\mathrm{r}}}{I_{\mathrm{z}}{v}_{\mathrm{x}}}}{\omega }_{\mathrm{z}}-{\displaystyle \frac{l_{\mathrm{a}}{k}_{\mathrm{f}}}{I_{\mathrm{z}}}}{\delta }_{\mathrm{f}}+{\displaystyle \frac{\Delta \mathrm{T}}{I_{\mathrm{z}}}}\end{array}\right.$$

(8)

Design of Yaw Rate Sliding Mode Control Algorithm

In the study of chassis control for self-propelled sprayers, the sliding mode control algorithm is widely considered due to its excellent robustness [30]. A sliding surface was designed using the actual and ideal yaw rates to further calculate the additional yaw rate moment T_ω.

The yaw rate tracking error and its derivative are defined using Equation (9) as follows:

$$\left\{\begin{array}{l}{e}_{\mathsf{\omega }}={\omega }_{\mathrm{z}}-{\omega }_{\mathrm{z}\_\exp }\\ {\dot{e}}_{\mathsf{\omega }}={\dot{\omega }}_{\mathrm{z}}-{\dot{\omega }}_{\mathrm{z}\_\exp }\end{array}\right.$$

(9)

The sliding mode surface s_ω is defined as follows:

$$s_{\mathsf{\omega }}=c_{\mathsf{\omega }}{e}_{\mathsf{\omega }}+{\dot{e}}_{\mathsf{\omega }}$$

(10)

where c_ω is the relative weighting coefficient between the error and the error rate of change; the sliding mode approach employs a constant rate reaching law; then, ${\dot{s}}_{\omega }=c_{\omega }{\dot{e}}_{\omega }+\stackrel{··}{e_{\omega }}=-{\epsilon }_{\omega }sign\left(s_{\omega }\right)$; ε_ω is a positive constant that determines the reaching speed; sign(s_ω) is the sign function.

By combining Equations (8)–(10), we obtain the following:

$${\dot{s}}_{\mathsf{\omega }}{=\mathrm{c}}_{\mathsf{\omega }}{\dot{e}}_{\mathsf{\omega }}+{\displaystyle \frac{l_{\mathrm{a}}{k}_{\mathrm{f}}-l_{\mathrm{b}}{k}_{\mathrm{r}}}{I_{\mathrm{z}}}}{\dot{\beta }}_{\mathrm{centroid}}+{\displaystyle \frac{l_{\mathrm{a}}^2{k}_{\mathrm{f}}+l_{\mathrm{b}}^2{k}_{\mathrm{r}}}{I_{\mathrm{z}}{v}_{\mathrm{x}}}}{\dot{\omega }}_{\mathrm{z}}-{\displaystyle \frac{l_{\mathrm{a}}{k}_{\mathrm{f}}}{I_{\mathrm{z}}}}{\dot{\delta }}_{\mathrm{f}}+{\displaystyle \frac{\Delta T}{I_{\mathrm{z}}}}-{\ddot{\omega }}_{\mathrm{z}\_\exp }$$

(11)

The additional yaw rate torque T_ω is finally obtained as follows:

$$T_{\mathsf{\omega }}=\int \left\{-{\mathrm{I}}_{\mathrm{z}}\left(c_{\mathsf{\omega }}{\dot{e}}_{\mathsf{\omega }}+{\displaystyle \frac{l_{\mathrm{a}}{k}_{\mathrm{f}}-l_{\mathrm{b}}{k}_{\mathrm{r}}}{I_{\mathrm{z}}}}{\dot{\beta }}_{\mathrm{centroid}}+{\displaystyle \frac{l_{\mathrm{a}}^2{k}_{\mathrm{f}}+l_{\mathrm{b}}^2{k}_{\mathrm{r}}}{I_{\mathrm{z}}{v}_{\mathrm{x}}}}{\dot{\omega }}_{\mathrm{z}}-{\displaystyle \frac{l_{\mathrm{a}}{k}_{\mathrm{f}}}{I_{\mathrm{z}}}}{\dot{\delta }}_{\mathrm{f}}-{\ddot{\omega }}_{\mathrm{z}\_\exp }+{\epsilon }_{\mathsf{\omega }}\cdot sign\left(s_{\mathsf{\omega }}\right)\right)\right\}dt$$

(12)

Design of Centroid Slip-Angle Sliding Mode Control Algorithm

The sliding mode surface was designed using the actual and desired values of the centroid sideslip angle to further solve for the additional centroid sideslip angle torque T_β. The tracking error of the centroid sideslip angle and its derivative are defined using Equation (13) as follows:

$$\left\{\begin{array}{l}{e}_{\mathsf{\beta }}={\beta }_{\mathrm{centroid}}-{\beta }_{\mathrm{z}\_\exp }\\ {\dot{e}}_{\mathsf{\beta }}={\dot{\beta }}_{\mathrm{centroid}}-{\dot{\beta }}_{\mathrm{z}\_\exp }\end{array}\right.$$

(13)

The sliding mode surface s_β is defined as follows:

$$s_{\mathsf{\beta }}=c_{\mathsf{\beta }}{e}_{\mathsf{\beta }}+{\dot{e}}_{\mathsf{\beta }}$$

(14)

where c_β is the relative weighting coefficient between the error and the error rate of change, which is greater than 0; the sliding mode approach employs a constant rate reaching law; then, ${\dot{s}}_{\mathsf{\beta }}=c_{\mathsf{\beta }}{\dot{e}}_{\mathsf{\beta }}+\stackrel{··}{e_{\mathsf{\beta }}}=-{\epsilon }_{\mathsf{\beta }}sign\left(s_{\mathsf{\beta }}\right)$; ε_β is a positive constant that determines the reaching speed; sign(s_β) is the sign function.

By combining Equations (8) and (14), the additional yaw torque T_β is obtained as follows:

$$T_{\mathsf{\beta }}{=-\mathrm{I}}_{\mathrm{z}}\left({\displaystyle \frac{l_{\mathrm{a}}{k}_{\mathrm{f}}-l_{\mathrm{b}}{k}_{\mathrm{r}}}{I_{\mathrm{z}}}}{\beta }_{\mathrm{centroid}}+{\displaystyle \frac{l_{\mathrm{a}}^2{k}_{\mathrm{f}}+l_{\mathrm{b}}^2{k}_{\mathrm{r}}}{I_{\mathrm{z}}{v}_{\mathrm{x}}}}{\omega }_{\mathrm{z}}-{\displaystyle \frac{l_{\mathrm{a}}{k}_{\mathrm{f}}}{I_{\mathrm{z}}}}{\delta }_{\mathrm{f}}+{\displaystyle \frac{{\epsilon }_{\mathsf{\beta }}sign\left(s_{\mathsf{\beta }}\right)+c_{\mathsf{\beta }}{\dot{e}}_{\mathsf{\beta }}+\frac{k_{\mathrm{f}}+k_{\mathrm{r}}}{M{v}_{\mathrm{x}}}{\dot{\beta }}_{\mathrm{centroid}}-\frac{k_{\mathrm{f}}}{M{v}_{\mathrm{x}}}{\dot{\delta }}_{\mathrm{f}}-{\ddot{\beta }}_{\mathrm{z}\_\exp }}{\frac{l_{\mathrm{a}}{k}_{\mathrm{f}}-l_{\mathrm{b}}{k}_{\mathrm{r}}}{M{v}_{\mathrm{x}}^2}-1}}\right)$$

(15)

Design of Weighting Module

Considering the coupling relationship between the yaw rate and the centroid sideslip angle, the sliding mode control generated two yaw control torques, T_ω and T_β, based on the desired yaw rate and centroid sideslip angle. These torques could not simultaneously achieve their respective desired values, thus requiring a weighted control approach.

When the centroid sideslip angle was below the lower threshold, the yaw rate was controlled, and the yaw torque T_ω generated based on the desired yaw rate was used as the final output. When the centroid sideslip angle exceeded the upper threshold, the centroid sideslip angle was controlled, and the yaw torque T_β generated based on the desired centroid sideslip angle was used as the final output. When the centroid sideslip angle was between these thresholds, a combined control was employed. The formula for the additional yaw torque distribution coefficient η is as follows:

$$\eta =\left\{\begin{array}{c}0,{\displaystyle \frac{\left|B_1{\dot{\beta }}_{\mathrm{centroid}}+{\beta }_{\mathrm{centroid}}\right|}{B_2}}=0\\ {\displaystyle \frac{\left|B_1{\dot{\beta }}_{\mathrm{centroid}}+{\beta }_{\mathrm{centroid}}\right|}{B_2}},0<{\displaystyle \frac{\left|B_1{\dot{\beta }}_{\mathrm{centroid}}+{\beta }_{\mathrm{centroid}}\right|}{B_2}}<1\\ 1,{\displaystyle \frac{\left|B_1{\dot{\beta }}_{\mathrm{centroid}}+{\beta }_{\mathrm{centroid}}\right|}{B_2}}\ge 1\end{array}\right.$$

(16)

The final combined yaw torque T_Z is expressed as follows:

$$T_{\mathrm{z}}=\eta T_{\mathsf{\beta }}+\left(1-\eta \right)T_{\mathsf{\omega }}$$

(17)

Design of PID Control Algorithm

To intuitively demonstrate the accuracy of the sliding mode control algorithm, comparative experiments using the PID algorithm were conducted. The PID control has fewer parameters, which are easy to determine, and it does not require precise knowledge of the mathematical model, making it useful when the mechanism of complex systems is hard to understand.

The PID algorithm calculates the difference between the actual feedback value and the reference value. In this study, the controlled variables were the centroid sideslip angle and the yaw rate. The proportional, integral, and derivative operations of the error were summed to form the additional torque control T, which was then fed into the lower-level controller. The PID control algorithm is expressed as follows:

$$u(t)=K_{\mathrm{p}}e+K_{\mathrm{i}}\int edt+K_{\mathrm{d}}{\displaystyle \frac{de}{dt}}$$

(18)

In Equation (18), u(t) represents the output control quantity, K_p is the proportional coefficient, K_i is the integral coefficient, and K_d is the derivative coefficient.

The inputs to the PID controller were the actual yaw rate and the centroid sideslip angle. The reference yaw rate and the centroid sideslip angle were provided by a two-degree-of-freedom model. The errors were processed by two separate PID controllers to obtain the yaw rate and the centroid sideslip angle control torques. These torques were then combined through a torque distribution module to form the control torque.

2.5. Design of Lower-Level Controller

The lower-level controller is a torque distribution controller, which reasonably allocates the longitudinal driving torque, yaw adjustment torque, and slip control torque required for the stable operation of the sprayer to the four-hub motors [31]. This study employed both the average and optimal distribution methods for allocating torque to the wheels and compared the effects of different distribution methods on vehicle stability.

2.5.1. Average Distribution Control

The average distribution method allocates the driving force and adjustment torque equally to each driving wheel during vehicle operation. The distribution formula is as follows:

$$\left\{\begin{array}{l}{T}_{\mathrm{fl}}=T_{\mathrm{fl}\_\mathrm{d}}-{\displaystyle \frac{T}{4}}+\Delta T_{\mathrm{s}\_\mathrm{fl}}\\ T_{\mathrm{fr}}=T_{\mathrm{fr}\_\mathrm{d}}+{\displaystyle \frac{T}{4}}+\Delta T_{\mathrm{s}\_\mathrm{fr}}\\ T_{\mathrm{rl}}=T_{\mathrm{rl}\_\mathrm{d}}-{\displaystyle \frac{T}{4}}+\mathsf{\Delta }{T}_{\mathrm{s}\_\mathrm{rl}}\\ T_{\mathrm{rr}}=T_{\mathrm{rr}\_\mathrm{d}}+{\displaystyle \frac{T}{4}}+\mathsf{\Delta }{T}_{\mathrm{s}\_\mathrm{rr}}\end{array}\right.$$

(19)

In the above formula, T_fl, T_fr, T_rl, and T_rr represent the driving torques of each wheel. T_{fl_d}, T_{fr_d}, T_{rl_d}, and T_{rr_d} denote the longitudinal control torques for each wheel obtained by the speed-following controller. The total control torque is given by T_d = T_{fl_d} + T_{fr_d} + T_{rl_d} + T_{rr_d}, where $T$ represents the yaw adjustment torque obtained from the yaw moment controller, and ΔT_{s_fl}, ΔT_{s_fr}, ΔT_{s_rl}, and ΔT_{s_rr} represent the adjustment torques of the four wheels obtained from the slip ratio controllers.

2.5.2. Optimal Distribution Control

Under constrained conditions, the optimization of the full-wheel torque distribution was carried out by determining the appropriate optimization objectives from the perspective of tire adhesion efficiency. This served as an indicator of vehicle stability. The minimum load rate of the tire was selected as the objective function [32], expressed as follows:

$${\eta }_{\mathrm{i}}={\displaystyle \frac{\sqrt{F_{\mathrm{xi}}^2+F_{\mathrm{yi}}^2}}{{\mu }_{\mathrm{i}}{F}_{\mathrm{zi}}}}$$

(20)

In the above equation, F_xi represents the longitudinal force on the wheel, F_yi represents the lateral force on the wheel, F_zi represents the vertical load on the wheel, and μ_i is the road adhesion coefficient of the wheel.

In engineering practice, the lateral force of the tire is more challenging to control precisely compared to the longitudinal force, and it is influenced by various dynamic factors. In contrast, the longitudinal force can be controlled more directly by adjusting the torque of the hub motors. Therefore, the objective function can be simplified as follows:

$$\min J=\min {\displaystyle \sum _{i=1}^4\frac{F_{xi}^2}{{\left({\mu }_{\mathrm{i}}{F}_{\mathrm{zi}}\right)}^2}}=\min {\displaystyle \sum _{i=1}^4\frac{T_{\mathrm{i}}^2}{{\left({\mu }_{\mathrm{i}}{F}_{\mathrm{zi}}R\right)}^2}}$$

(21)

Considering the constraints of road surface friction characteristics and the maximum torque output of the motor, the vehicle was assumed to be symmetric, with the front axle track width equal to the rear axle track width. The boundary conditions were set as follows:

$$\left\{\begin{array}{l}{T}_{\mathrm{fl}}\cos {\delta }_{\mathrm{f}}+T_{\mathrm{fr}}\cos {\delta }_{\mathrm{f}}+T_{\mathrm{rl}}+T_{\mathrm{rr}}=T_{\mathrm{d}}\\ {\displaystyle \frac{B}{2R}}\left(T_{\mathrm{fl}}+T_{\mathrm{fr}}\right)\cos {\delta }_{\mathrm{f}}+{\displaystyle \frac{B}{2R}}\left(T_{\mathrm{rl}}+T_{\mathrm{rr}}\right)=\Delta T\\ T_{\mathrm{fij}}\le \min (\mu F_{\mathrm{zij}}R,T_{\max })\end{array}\right.$$

(22)

In the above equation, T_fij represents the torque of each wheel, F_zij denotes the vertical load on the wheel, T_max is the maximum output torque of the hub motor, and B is the track width of the sprayer.

By substituting the equality constraints into the objective function, the following is obtained:

$$\left\{\begin{array}{l}{T}_{\mathrm{fl}}={\displaystyle \frac{1}{\cos {\delta }_{\mathrm{f}}}}\left({\displaystyle \frac{T_{\mathrm{d}}}{2}}-{\displaystyle \frac{\Delta T}{B}}R-T_{\mathrm{rl}}\right)\\ T_{\mathrm{fr}}={\displaystyle \frac{1}{\cos {\delta }_{\mathrm{f}}}}\left({\displaystyle \frac{T_{\mathrm{d}}}{2}}+{\displaystyle \frac{\Delta T}{B}}R-T_{\mathrm{rr}}\right)\end{array}\right.$$

(23)

Substituting Equation (23) into Equation (21) results in a new objective function as follows:

$$J={\displaystyle \frac{{\left(\frac{T_{\mathrm{d}}}{2}-\frac{\Delta T}{B}R-T_{\mathrm{rl}}\right)}^2}{{\left(\cos {\delta }_{\mathrm{f}}{\mu }_{\mathrm{fl}}{F}_{\mathrm{zfl}}R\right)}^2}}+{\displaystyle \frac{{\left(\frac{T_{\mathrm{d}}}{2}+\frac{\Delta T}{B}R-T_{\mathrm{rr}}\right)}^2}{{\left(\cos {\delta }_{\mathrm{f}}{\mu }_{\mathrm{fr}}{F}_{\mathrm{zfr}}R\right)}^2}}+{\displaystyle \frac{T_{\mathrm{rl}}^2}{{\left({\mu }_{\mathrm{rl}}{F}_{\mathrm{zrl}}R\right)}^2}}+{\displaystyle \frac{T_{\mathrm{rr}}^2}{{\left({\mu }_{\mathrm{rr}}{F}_{\mathrm{zrr}}R\right)}^2}}$$

(24)

By taking the partial derivatives of the above equation,

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial J}{\partial T_{\mathrm{rl}}}}=-{\displaystyle \frac{2\left(\frac{T_{\mathrm{d}}}{2}-\frac{\Delta T}{B}R-T_{\mathrm{rl}}\right)}{{\left(\cos {\delta }_{\mathrm{f}}{\mu }_{\mathrm{f}l}{F}_{{\mathrm{z}}_{\mathrm{fl}}}R\right)}^2}}+{\displaystyle \frac{2T_{\mathrm{rl}}}{{\left({\mu }_{\mathrm{rl}}{F}_{{\mathrm{z}}_{\mathrm{rl}}}R\right)}^2}}\\ {\displaystyle \frac{\partial J}{\partial T_{\mathrm{rr}}}}=-{\displaystyle \frac{2\left(\frac{T_{\mathrm{d}}}{2}+\frac{\Delta T}{B}R-T_{\mathrm{rr}}\right)}{{\left(\cos {\delta }_{\mathrm{f}}{\mu }_{\mathrm{fr}}{F}_{{\mathrm{z}}_{\mathrm{fr}}}R\right)}^2}}+{\displaystyle \frac{2T_{\mathrm{rr}}}{{\left({\mu }_{\mathrm{rr}}{F}_{{\mathrm{z}}_{\mathrm{rr}}}R\right)}^2}}\end{array}\right.$$

(25)

Setting $\partial$J/$\partial$T_rl and $\partial$J/$\partial$T_rr in Equation (25) to zero yields T_rl and T_rr as follows:

$$\left\{\begin{array}{l}{T}_{\mathrm{rl}}={\displaystyle \frac{\frac{{\mu }_{\mathrm{rl}}^2{F}_{\mathrm{zrl}}^2{T}_{\mathrm{d}}}{2}-\frac{{\mu }_{\mathrm{rl}}^2{F}_{\mathrm{zrl}}^2\Delta T}{B}R}{{\mu }_{\mathrm{rl}}^2{F}_{\mathrm{zrl}}^2+{\mu }_{\mathrm{f}l}^2{F}_{\mathrm{zfl}}^2{\cos }^2{\delta }_{\mathrm{f}}}}\\ T_{\mathrm{rr}}={\displaystyle \frac{\frac{{\mu }_{\mathrm{rr}}^2{F}_{\mathrm{zrr}}^2{T}_{\mathrm{d}}}{2}+\frac{{\mu }_{\mathrm{rr}}^2{F}_{\mathrm{zrr}}^2\Delta T}{B}R}{{\mu }_{\mathrm{rr}}^2{F}_{\mathrm{zrr}}^2+{\mu }_{\mathrm{fr}}^2{F}_{\mathrm{zfr}}^2{\cos }^2{\delta }_{\mathrm{f}}}} \end{array}\right.$$

(26)

Substituting these results into Equation (23) allows for the determination of T_fl and T_fr.

3. Results and Analysis

3.1. Controller Integration

To validate the effectiveness of the stability controller, we conducted simulation experiments in Simulink focusing on two scenarios: serpentine driving during the relocation of the sprayer for transportation and low-speed steering under field operating conditions. These experiments were carried out using MATLAB version R2021b, where the primary construction of function bodies was achieved through the use of the function module to process input parameters and perform related functions. In these tests, the combined effects of two control methods were compared against a baseline without control.

The first combination involved the upper-level controller as the speed-following controller, the anti-slip fuzzy PID controller, and the direct yaw moment PID controller. The lower-level controller employed the average torque distribution control, forming the integrated controller shown in Figure 10. The proportional, integral, and derivative coefficients of the upper-level PID controller were 80, 2, and 0.1, respectively.

The second combination used the upper-level controller as the speed-following controller, the anti-slip fuzzy PID controller, and the direct yaw moment sliding mode controller. The lower-level controller employed the optimal torque distribution control, as shown in Figure 11. The sliding mode control parameters for the yaw rate were c_ω = 3 and ε_ω = 5, while those for the sideslip angle of the vehicle’s center of mass were c_β = 2 and ε_β = 5.

3.2. Transition Transport Conditions

3.2.1. Straight-Line Acceleration Scenarios (High Adhesion Pavement)

The adhesion coefficient of the road surface while the sprayer was performing linear acceleration on the pavement was set at 0.8, and its speed was increased from 5 km/h to the designated desired vehicle speed of 20 km/h. In Simulink, the simulation time was set to 10 s with a simulation step size of 0.001 s. The simulation results are shown in Figure 12a–d, and representative parameters of the simulation results are summarized in

Table 5. Comparison of parameter indexes under the condition of transfer field transportation–straight line acceleration.

Evaluation Indicators	No Control	PID + Average Distribution	SMC + Optimal Distribution
Steady-state longitudinal velocity error rate/-	2.56%	1.17%	1.19%
Front-wheel peak slip rate/%	43.61	17.60	16.62
Rear-wheel peak slip rate/%	40.63	17.06	17.35
Trajectory deviation/m	−2.38	0.00	0.00

.

The analysis results indicate that under high-adhesion road conditions, the control system effectively determines wheel slip rates during straight-line acceleration. As shown in Figure 12a, the sprayer’s speed increases smoothly and steadily without overshooting after the introduction of the anti-slip control module. Figure 12b shows that without control, the sprayer’s trajectory deviation is significant, approaching 2.5 m at the end of the 10 s simulation. With control, the deviation is nearly eliminated, ensuring effective straight-line driving. The analysis of Figure 12c,d and Table 5 reveal that without control, the slip rates of both the front and rear wheels exceed 0.4, indicating severe wheel slip. Using the “SMC + optimal distribution” method reduces the peak slip rate of the front wheels to 0.1662, while the “PID + average distribution” control method is slightly better for the rear wheels, controlling the peak slip rate at 0.1706.

3.2.2. Snake-Shaped Driving Scenarios

The sprayer was set to perform a serpentine maneuver on a high-adhesion road with a coefficient of 0.8 at a speed of 15 km/h. The simulation time was set to 10 s with a step size of 0.001 s. Steering angle control signals, as shown in Figure 13a, were applied to the sprayer’s front wheels. Comparative analysis was conducted for three scenarios: no stability control, PID + average distribution control, and SMC + optimal distribution control. Figure 13b–d display the state parameters of the sprayer under different control methods, and

Table 6. Comparison of evaluation indexes under transfer field transportation–snake driving.

Evaluation Indicators	No Control	PID + Average Distribution	SMC + Optimal Distribution
Maximum yaw rate deviation/rad·s−1	0.049	0.037	0.006
Maximum deviation of the centroid sideslip angle/rad	0.058	0.055	0.048
Maximum longitudinal coordinate offset/m	1.020	0.620	0.280

shows a comparison of the evaluation metrics of the sprayer.

The analysis of Figure 13 and Table 6 reveals that both the “PID + average distribution” and “SMC + optimal distribution” control methods effectively suppress sprayer instability. However, the “SMC + optimal distribution” control method improves the control of the yaw rate and centroid sideslip angle by 83.7% and 12.7%, respectively, compared to the “PID + average distribution” method. Additionally, this control method more effectively follows the reference path, reducing the maximum longitudinal coordinate offset by 54.8% compared to the “PID + average distribution” method, with only a 2.03% deviation from the reference value, thus ensuring excellent trajectory-following performance during transit.

3.3. Fieldwork Conditions

3.3.1. Straight-Line Acceleration Scenarios (Low Adhesion Pavement)

Assuming the current road surface has low adhesion, the coefficient of adhesion was set to 0.3. The speed of the sprayer during spraying was 5 km/h, and the vehicle was simulated to accelerate from 0 to the set desired speed of 5 km/h. The simulation duration was set to 10 s, and the simulation time step was 0.001 s. Simultaneously, during the simulation, a signal was input that gradually reduced the sprayer’s ready-to-use mass at a rate of 1 kg/s, to better simulate the actual spraying situation of the sprayer. The simulation results are shown in Figure 14a–d, and representative parameters of the simulation results are summarized in

Table 7. Comparison of parameter indexes under field operation—straight-line acceleration condition.

Evaluation Indicators	No Control	PID + Average Distribution	SMC + Optimal Distribution
Steady-state longitudinal velocity error rate/-	8.97%	1.94%	0.12%
Front-wheel peak slip rate/%	14.76	9.69	8.40
Rear-wheel peak slip rate/%	24.86	6.48	5.65
Trajectory deviation/m	−0.38	0.00	0.00

.

Analysis of Figure 14 and Table 7 reveals that both “PID + average distribution” and “SMC + optimal distribution” control methods achieved the desired speed by the 7th second, with final steady-state speed errors less than 2%. Without control, the wheel slip ratio was the highest, particularly for the rear wheels, where the peak value was 0.2486. Through control, the peak values were significantly reduced to 0.0648 and 0.0565, respectively. Compared to the transit scenario, the peak value for the front wheels without control was 0.1476, which was within a reasonable range. After control, it was reduced to less than 0.1. This is because the expected speed during field operations is lower, resulting in smaller longitudinal acceleration and, consequently, less wheel slip.

3.3.2. Slow-Turn Driving Scenarios

A step signal from 0 to 0.1 was applied to the front axle to simulate the stability of the sprayer during emergency obstacle avoidance in its operation. Simultaneously, the sprayer was set to steer on a road surface with low adhesion, characterized by a coefficient of 0.3, while the vehicle maintained a constant speed of 5 km/h. The simulation duration was set to 10 s, with a time step of 0.001 s for the simulation. The simulation results are shown in Figure 15a–d, and representative parameters of the simulation results are summarized in

Table 8. Comparison of evaluation indexes under field operation—low-speed steering condition.

Evaluation Indicators	No Control	PID + Average Distribution	SMC + Optimal Distribution
Maximum yaw rate deviation/rad·s−1	0.035	0.021	0.003
Maximum deviation of the centroid sideslip angle/rad	0.060	0.057	0.054
Maximum longitudinal coordinate offset/m	4.585	0.304	0.165

.

The analysis of Figure 15 and Table 8 indicates that the sprayer’s low-speed steering performance without control significantly deviates from the ideal steering characteristics, resulting in instability. The ideal yaw rate shows a step change at 2 s. The “SMC + optimal distribution” control method follows the ideal value most closely, with a maximum deviation of only 0.003, improving control by 85.7% compared to the “PID + average distribution” control. Additionally, the “SMC + optimal distribution” control method provides the best tracking of the centroid sideslip angle, with the smallest deviation from the desired trajectory, demonstrating superior control effectiveness.

4. Discussion and Prospect

(1) In this study, we performed a comparative analysis of the “PID + average distribution” control and “SMC + optimal distribution” control against a no-control baseline to evaluate their impact on the stability of the spray machine’s chassis. The findings indicate that under scenarios of linear acceleration, serpentine maneuvers, and low-speed turns, the spray machine’s chassis stability was significantly enhanced with the application of control strategies compared to the no-control condition, resulting in improved maneuverability and safety.

In particular, during direct acceleration in both transit and field operation conditions, the steady-state longitudinal speed error rate averaged 0.655% under the “SMC + optimal distribution” control, a 0.9% reduction compared to the “PID + average distribution” control. Both control approaches yielded similar outcomes in wheel slip rate regulation. Under serpentine and low-speed turning scenarios, the “SMC + optimal distribution” control resulted in an average peak yaw angular velocity deviation of 0.0045 and an average peak center of gravity lateral offset deviation of 0.051, representing reductions of 0.0245 and 0.005, respectively, over the “PID + average distribution” control.

Thus, the “SMC + optimal distribution” control method notably decreased the yaw angular velocity and center of gravity lateral offset deviation during steering and linear driving while maintaining the wheel slip rate within an optimal range, thereby directly enhancing the spray vehicle’s driving stability. Moreover, this integrated control method enhances adaptability across various operational conditions, ensuring superior stability control in diverse environmental settings.

(2) Taking into account the significance of the suspension system for chassis stability, in subsequent research, the suspension model can be introduced to enhance the accuracy of the sprayer’s dynamic model, allowing for a deeper analysis and optimization of its role in chassis stability control. Furthermore, considering variations in terrain types, weather conditions, and crop types, different terrains and weather conditions lead to discrepancies in ground friction coefficients, impacting wheel slip rates. This necessitates more frequent and refined adjustments by the control system. Moreover, diverse crop types impose different requirements on field operations, requiring control strategies to take these factors into account to ensure efficient and stable spraying operations without damaging the crops.

(3) In future research, more complex models and sensor data fusion can be introduced to enhance the control system’s capability to perceive environmental changes, enabling adaptive control under various conditions. Simultaneously, real vehicle tests should be conducted in a wide range of practical environments to collect performance data of the control system under different conditions, thereby evaluating its robustness and adaptability.

5. Conclusions

(1) This study established an ideal model based on the seven-degree-of-freedom model of the high-clearance self-propelled electric sprayer. The stability of the sprayer’s body was assessed using an instability judgment module. An upper-layer stability controller and a lower-layer torque distribution controller were designed to control the centroid sideslip angle, yaw rate, and wheel slip ratio during operation, achieving stability control of the sprayer.

(2) A hierarchical stability control framework was adopted, and simulation experiments were conducted under four operating conditions: transit transport and field operations. The control strategies of “SMC + optimal distribution” and “PID + average distribution” control were compared with no control to analyze the stability characteristics under different control strategies. The simulation results demonstrate that the designed control algorithms optimized the stability parameters of the sprayer to varying degrees, improving the stability control of the high-clearance self-propelled electric sprayer with a four-wheel independent drive.

(3) This paper advances the theoretical understanding of stability control for self-propelled sprayers, specifically in the areas of nonlinear model development, control strategy design, and simulation validation. By introducing fuzzy PID theory, SMC theory, and optimal torque distribution algorithms, it offers innovative technical approaches to stabilizing sprayers. These advancements hold significant technological value in the current trend toward the intelligentization and automation of agricultural machinery. In terms of practical application, the results of simulation trials substantiate the efficacy of the control strategies, providing a basis for the optimization and enhancement of self-propelled sprayer design and performance. This contributes to improved efficiency and safety in agricultural operations.